Due on October 9, 2006, Monday, Class time. No late submissions!

MATH 302 Homework 1

1: Evaluate

∫ ∞

0

dx

1 + xαwhere α > 1. Take into account that zα = exp(α ln z) is not

defined at the origin.

2: Evaluate

∫ ∞

0

dx

(1 + x2)nwhere n ≥ 1 is an integer.

3: Find a conformal mapping of the disc x2 + (y − 1)2 < 1 onto the first quadrantx, y > 0. Investigate the conformal property of your map also on the boundaries.

4: Describe the image of the unit disc under the transformation ln

(z − 1

z + 1

), where an

appropriate branch of the logarithm is used.